4.5 Applications in the Numerical Analyses
93
Fig. 4.11 HJC model a equation of yield surface, b equation of state c damage model, reprinted
from Ren et al. (2016), copyright 2020, with permission from Elsevier
Shown in Fig. 4.11b, the pressure-volumetric strain relationship is separated into
three response regions in the equation of state of HJC model. It is expressed as
⎧
⎨
⎩
P = Kμ
P < P crush
P = P crush + K lock (μ − μ crush ) P crush < P < P lock
P = K 1 ¯
μ + K 2 ¯
μ
2
+ K 3 ¯
μ
3
P > P lock
(4.11)
where μ = ρ/ρ 0 − 1 is the volumetric strain, ρ and ρ 0 are the current and initial
density, respectively. The elastic bulk modulus is K = P crush /μ crush , where P crush
and μ crush are the pressure and volumetric strain at which the material begins to
undergo plastic deformation. K lock = (P lock − P crush )/(μ plock − μ crush ), and μ plock
is the volumetric strain at the compacting hydrostatic pressure P lock . μ = (μ −
μ lock )/(1 + μ lock ) is the modified volumetric strain, and the constants K 1 , K 2 and
K 3 are equivalent to those used for material with no voids.
93
Fig. 4.11 HJC model a equation of yield surface, b equation of state c damage model, reprinted
from Ren et al. (2016), copyright 2020, with permission from Elsevier
Shown in Fig. 4.11b, the pressure-volumetric strain relationship is separated into
three response regions in the equation of state of HJC model. It is expressed as
⎧
⎨
⎩
P = Kμ
P < P crush
P = P crush + K lock (μ − μ crush ) P crush < P < P lock
P = K 1 ¯
μ + K 2 ¯
μ
2
+ K 3 ¯
μ
3
P > P lock
(4.11)
where μ = ρ/ρ 0 − 1 is the volumetric strain, ρ and ρ 0 are the current and initial
density, respectively. The elastic bulk modulus is K = P crush /μ crush , where P crush
and μ crush are the pressure and volumetric strain at which the material begins to
undergo plastic deformation. K lock = (P lock − P crush )/(μ plock − μ crush ), and μ plock
is the volumetric strain at the compacting hydrostatic pressure P lock . μ = (μ −
μ lock )/(1 + μ lock ) is the modified volumetric strain, and the constants K 1 , K 2 and
K 3 are equivalent to those used for material with no voids.